Lectures on orbifolds and reflection groups Michael W. Davis * July 23, 2008 These are the notes for my lectures in the Summer School on Transfor- mations Groups and Orbifolds held at the CMS of Zhejiang University in Hangzhou, China from June 30 to July 11, 2008. The notes closely follow the slides which I used to present my lectures. Most of the material in the first four lectures comes from parts of Bill Thurston’s 1976-77 course at Princeton University. Although this material has not been published, it can be found in [13] at the given electronic address. Contents 1 Lecture 1: transformation groups and orbifolds 2 1.1 Transformation groups ...................... 2 1.1.1 Definitions ......................... 2 1.1.2 The Differentiable Slice Theorem ............ 3 1.1.3 Proper actions of discrete groups ............ 4 1.2 Orbifolds .............................. 4 1.2.1 Definitions and terminology ............... 4 1.2.2 Covering spaces and π orb 1 ................. 7 1.2.3 1- and 2-dimensional orbifolds .............. 8 1.2.4 General orbifolds ..................... 10 1.3 Generators and relations for π orb 1 (Q) ............... 12 2 Lecture 2: two-dimensional orbifolds 12 2.1 Orbifold Euler characteristics .................. 12 2.2 Classification of 2-dimensional orbifolds ............. 14 * The author was partially supported by NSF grant DMS 0706259. 1
Transcript
Lectures on orbifolds and reflection groups
Michael W. Davis∗
July 23, 2008
These are the notes for my lectures in the Summer School on Transfor-mations Groups and Orbifolds held at the CMS of Zhejiang University inHangzhou, China from June 30 to July 11, 2008. The notes closely followthe slides which I used to present my lectures.
Most of the material in the first four lectures comes from parts of BillThurston’s 1976-77 course at Princeton University. Although this materialhas not been published, it can be found in [13] at the given electronic address.
An action of a topological group G on a space X is a (continuous) mapG×X → X, denoted by (g, x)→ gx, so that
• g(hx)=(gh)x,
• 1x=x.
(Write G y X to mean that G acts on X.)Given g ∈ G, define θg : X → X by x → gx. Since θg ◦ θg−1 = 1X =
θg−1 ◦ θg, the map θg is a homeomorphism and the map Θ : G→ Homeo(X)defined by g → θg is a homomorphism of groups.
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Given x ∈ X, Gx := {g ∈ G | gx = x} is the isotropy subgroup. Theaction is free if Gx = {1}, for all x ∈ X.
Definitions 1.1. G(x) := {gx ∈ X | g ∈ G} is the orbit of x. The action istransitive if there is only one orbit. Given x ∈ X, the natural map G/Gx →G(x) defined by gGx → gx is a continuous bijection. The orbit space X/G isthe set of orbits in X endowed with the quotient topology (with respect tothe natural map X → X/G). A map f : X → Y of G-spaces is equivariant(or a G-map) if f(gx) = gf(x)
Definitions 1.2. Suppose H ⊂ G is a subgroup and Y is a H-space. ThenH acts on G × Y via h · (g, x) = (gh−1, hx). The orbit space is denotedG ×H Y and called the twisted product. The image of (g, x) in G ×H Y isdenoted [g, x]. Note that G y G×H Y via g′[g, x] = [g′g, x].
Definition 1.3. A slice at a point x ∈ X is a Gx-stable subset Ux so that themap G×Gx Ux → X is an equivariant homeomorphism onto a neighborhoodof G(x). If Ux is homeomorphic to a disk, then G ×Gx Ux is an equivarianttubular neighborhood of G(x).
Remark 1.4. A neighborhood of the orbit in X/G is homeomorphic toUx/Gx (= (G×Gx Ux)/G).
1.1.2 The Differentiable Slice Theorem
The next result is basic in the study of smooth actions of compact Lie groups(including finite groups) on manifolds. For details, see [3].
Theorem 1.5. Suppose a compact Lie group acts differentiably (= “smoothly”)on a manifold M . Then every orbit has a G-invariant tubular neighborhood.More precisely, there is a linear representation of Gx on a vector space S sothat that G×Gx S is a tubular neighborhood of G(x). (The image of S in Mis a slice at x.)
Proof. By integrating over the compact Lie group G we can find a G-invariantRiemannian metric. Then apply the usual proof using the exponential map.
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1.1.3 Proper actions of discrete groups
Suppose Γ a discrete group, X a Hausdorff space and Γ y X. The Γ-actonis proper if given any two points x, y ∈ X, there are open neighborhoods Uof x and V of y so that γU ∩ V 6= ∅ for only finitely many γ.
Exercise 1.6. Show that a Γ-action on X is proper iff
• X/Γ is Hausdorff,
• each isotropy subgroup is finite,
• each point x ∈ X has a slice, i.e., there is Γx-stable open neighborhoodUx so that γUx ∩ Ux = ∅, for all γ ∈ Γ − Γx. (This means thatΓ×Γx Ux maps homeomorphically onto a neighborhood of the orbit ofx.)
Actions on manifolds. Suppose a discrete group Γ acts properly on an n-dimensionalt manifold Mn. A slice Ux at x ∈Mn is linear if there is a linearΓx-action on Rn so that Ux is Γx-equivariantly homeomorphic to a Γx-stableneighborhood of the origin in Rn. The action is locally linear if every pointhas a linear slice.
Proposition 1.7. If Γ y Mn properly and differentiablly, then action islocally linear.
Proof. Since Γx is finite, we can find a Γx-invariant Riemannian metric onM . The exponential map, exp : TxM → M , is Γx-equivariant and takes asmall open disk about the origin homeomorphically onto a neighborhood Uxof x. If the disk is small enough, Ux is a slice.
1.2 Orbifolds
1.2.1 Definitions and terminology
Definition 1.8. An orbifold chart on a space X is a 4-tuple (U , G, U, π),where
• U is open subset of X,
• U is open in Rn and G is finite group of homeomorphisms of U ,
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• π : U → U is a map which can be factored as π = π ◦ p, wherep : U → U/G is the orbit map and π : U/G→ U is a homeomorphism.
The chart is linear if the G-action on Rn is linear.
For i = 1, 2, suppose (Ui, Gi, Ui, πi) is an orbifold chart on X. The charts
are compatible if given points ui ∈ Ui with π1(u1) = π2(u2), there is a home-
omorphism h from neighborhood of u1 in U1 onto neighborhood of u2 in U2
so that π1 = π2 ◦ h on this neighborhood.
Definition 1.9. An orbifold atlas on X is a collection {(Ui, Gi, Ui, πi)}ı∈Iof compatible orbifold charts which cover X. An orbifold Q consists of anunderlying space |Q| together with an atlas of charts.
An orbifold is smooth if the groups act via diffeomorphisms and the chartsare compatible via diffeomorphisms. A locally linear orbifold means all chartsare equivalent to linear ones. By the Differentiable Slice Theorem a smoothorbifold is locally linear.
From now on, all orbifolds will be locally linear
Exercise 1.10. Suppose Γ acts properly on a manifold Mn. By choosingslices we can cover M/Γ with compatible orbifold charts. Show this gives theunderlying space M/Γ the structure of orbifold, which we denote by M//Γ.
Remark 1.11. (Groupoids). As Professors Adem and Xu said in their talks,the best way to view an orbifold is as a groupoid. This point was first madeby Haefliger [9]. Given an atlas {(Ui, Gi, Ui, πi)}ı∈I for an orbifold Q oneassociates a groupoid G to it as follows. The set of object G0 is the disjointunion:
G0 :=∐i∈I
Ui.
The set of morphisms G1 is defined as follows. Given ui ∈ Ui and uj ∈ Uj,a morphism ui → uj is the germ of a local homeomorphism U → V froma neighborhood of ui to a neighborhood of uj which commutes with theprojections, πi and πj. (Note: in the above we can take i = j and f to bethe germ of translation by a nontrivial element γ ∈ Gi.)
The local group There is more information in an orbifold than just itsunderlying space. For example, if q ∈ |Q| and x ∈ π−1(q) is a point in
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the inverse image of q in some local chart, then the isotropy subgroup Gx
is independent of the chart, up to an isomorphism of groups. With thisambiguity, we call it the local group at q and denote it by Gq.
A manifold is an orbifold in which each local group is trivial.
Strata. In transformation groups, if G y X and H ⊂ G, then
X(H) := {x ∈ X | Gx is conjugate to H}
is the set of points of orbit type G/H. The image of X(H) in X/G is a stratumof X/G.
This image can be described as follows. First, take the fixed set XH
(:= {x ∈ X | hx = x,∀h ∈ H}). Next, remove the points x with Gx ) Hto get XH
(H). Then divide by the free action of N(H)/H to get X∗(H), the
stratum of type (H) in X/G. In an orbifold, Q, a stratum of type (H) is thesubspace of |Q| consisting of all points with local group isomorphic to H.
Proposition 1.12. If Q is a locally linear orbifold, then each stratum is amanifold.
Proof. Suppose a finite group G y Rn linearly and H ⊂ G. Then (Rn)H isa linear subspace; hence, (Rn)H(H) is a manifold. Dividing by the free action
of N(H)/H, we see that (Rn)∗(H) is a manifold.
The origin of the word “orbifold”: the true story. Near the beginningof his graduate course in 1976, Bill Thurston wanted to introduce a word toreplace Satake’s “V-manifold” from [12]. His first choice was “manifolded”.This turned out not to work for talking - the word could not be distinguishedfrom “manifold”. His next idea was “foldimani”. People didn’t like this. SoBill said we would have an election after people made various suggestionsfor a new name for this concept. Chuck Giffen suggested “origam”, DennisSullivan “spatial dollop” and Bill Browder “orbifold”. There were manyother suggestions. The election had several rounds with the names havingthe lowest number of votes being eliminated. Finally, there were only 4 namesleft, origam, orbifold, foldimani and one other (maybe “V-manifold”). Afterthe next round of voting “orbifold” and the other name were to be eliminated.At this point, I spoke up and said something like “Wait you can’t eliminateorbifold because the other two names are ridiculous.” So “orbifold” was lefton the list. After my impassioned speech, it won easily in the next round ofvoting.
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1.2.2 Covering spaces and πorb1
Thurston’s big improvement over Satake’s earlier versionin [12] was to showthat the theory of covering spaces and fundamental groups worked for orb-ifolds. (When I was a graduate student a few years before, this was “well-known” not to work.)
The local model for a covering projection between n-dimensional mani-folds is the identity map, id : U → U , on an open subset U ⊂ Rn. Simi-larly, the local model for an orbifold covering projection is the natural mapRn/H → Rn/G where a finite group G y Rn and H ⊂ G is a subgroup.
Proposition 1.13. If Γ acts properly onM and Γ′ ⊂ Γ is a subgroup, thenM//Γ′ →M//Γ is an orbifold covering projection.
Definition 1.14. An orbifold Q is developable if it is covered by a manifold.As we will see, this is equivalent to the condition that Q be the quotient ofa discrete group acting properly on a manifold. (In Thurston’s terminology,Q is a “good” orbifold.)
Remark 1.15. Not every orbifold is developable (later we will describe the“tear drop,” the standard counterexample).
Definition 1.16. Q is simply connected if it is connected and does not admita nontrivial orbifold covering, i.e., if p : Q′ → Q is a covering with |Q′|connected, then p is a homeomorphism.
Fact. Any connected orbifold Q admits a simply connected orbifold coveringπ : Q → Q. This has the usual universal property: if we pick a “generic”base point q ∈ Q and p : Q′ → Q is another covering with base points q′ ∈ Q′and q ∈ Q lying over q, then π factors through Q′ via a covering projectionQ→ Q′ taking q to q′. In particular, Q→ Q is a regular covering in the sensethat its group of deck transformations acts simply transitively on π−1(q). (Asimply transitive action is one which is both free and transitive.)
Definitions of the orbifold fundamental group.
Definition 1.17. (cf. [13]). πorb1 (Q) is the group of deck transformations of
the universal orbifold cover, p : Q→ Q
There are three other equivalent definitons of πorb1 (Q), which we list below.Each involves some technical difficulties.
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• In Subsection 1.3, I will give a definition in terms of generators andrelations.
• A third definition is in terms of “homotopy classes” of “loops” [0, 1]→Q. The difficulty with this approach is that we must first define whatis meant by a “map” from a topological space to Q - it should be acontinuous map to |Q| together with a choice of a “local lift” (up toequivalence) for each orbifold chart for Q.
• A fourth definition is in terms of the groupoid. If GQ is the groupoidassociated to Q and BGQ is its classifying space, then πorb1 (Q) :=π1(BGQ), the ordinary fundamental group of the space BGQ. The onlyproblem with this definition is that one first needs to define the classi-fying space of a groupoid.
Developability and the local group. For each x ∈ |Q|, let Gx denote thelocal group at x. (It is a finte subgroup of GL(n,R), well-defined up to con-jugation. We can identify Gx with the fundamental group of a neighborhoodof the form Ux/Gx where Ux is a ball in some linear representation. So, Gx
is the “local fundamental group” at x. The inclusion of the neighborhoodinduces a homomorphism Gx → πorb1 (Q).
Proposition 1.18. Q is developable ⇐⇒ each local group injects (i.e., foreach x ∈ |Q|, the map Gx → πorb1 (Q) is injective).
1.2.3 1- and 2-dimensional orbifolds
Dimension 1. The only finite group which acts linearly (and effectively) onR1 is the cyclic group of order 2, C2. It acts via the reflection x 7→ −x. Theorbit space R1/C2 is identified with [0,∞).
It follows that every 1-dimensional orbifold Q is either a 1-manifold or a1-manifold with boundary. If Q is compact and connected, then it is eithera circle or an interval (say, [0, 1]).
The infinite dihedral group, D∞ is the group generated by 2 distinctaffine reflections on R1 and R1/D∞ ∼= [0, 1]. (See figure 1.) It follows thatthe universal orbifold cover of [0, 1] is R1.
2-dimensional linear groups. Suppose a finite group G y Rn linearly.Then G is conjugate to subgroup of O(n). (Pf: By averaging we get aninvariant inner product). Hence, G acts on the unit sphere Sn−1 ⊂ Rn.
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r r´
-1 0 1
Figure 1: The infinite dihedral group
Suppose G ⊂ O(2). Then S1//G = S1 or S1//G = [0, 1].
• In the first case, S1 → S1//G = S1 is an n-fold cover, where n = |G|,and G is the cyclic group Cn acting by rotations.
• In the second case, the composition, R1 → S1 → S1//G = [0, 1], isthe universal orbifold cover with group of deck transformations D∞.It follows that G = Dm (the dihedral group of order 2m) or G = C2
(= D1) acting by reflection across a line.
Theorem 1.19. (Theorem of Leonardo da Vinci, cf. [14, pp. 66, 99].) Anyfinite subgroup of O(2) is conjugate to either Cn or Dm.
rL
rL´
π/m
Question. What does R2//G look like?
Here are the possibilities:
• R2 (G = {1}),
• a cone (G = Cn),
• a half-space (G = D1),
• a sector (G = Dm).
In the half-space case, a codimension 1 stratum is a mirror. In the sectorcase, a codimension 2 stratum is a corner reflector.
2-dimensional orbifolds. Here is the picture: the underlying space of a2-dimensional orbifold Q is a 2-manifold, possibly with boundary . Certain
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Figure 2: Not developable
points in the interior of the |Q| are “cone points” labeled by an integer nispecifying that the local group is Cni
. The codimension 1 strata are themirrors; their closures cover ∂|Q|. The closures of two mirrors intersect ina corner reflectors (where local group is Dmi
). The picture in Figure 2 ispossible; however, it is not developable.
1.2.4 General orbifolds
• If G ⊂ O(n) and Dn ⊂ Rn denotes the unit disk, then G y Dn.
• Since Dn = Cone(Sn−1), we have Dn//G = Cone(Sn−1//G). Therefore,a point in a general orbifold has a conical neighborhood of this form.
Example 1.20. Suppose G = C2 acting via antipodal map, x 7→ −x. ThenDn//C2 = Cone(RP n−1)
Suppose Q is an n-dimensional orbifold and Q(2) denotes the complementof the strata of codimension > 2. The description of Q(2) is similar to a 2-dimensional orbifold. |Q(2)| is an n-manifold with boundary ; the boundaryis a union of (closures of) mirrors; the codimension 2 strata in the interiorare codimension 2 submanifolds labeled by cyclic groups; the codimension 2strata on the boundary are corner reflectors labeled by dihedral groups.
During one of the problem sessions I was asked the following question.
Question. When is the underlying space of an orbifold a manifold?
This question is equivalent to the following.
Question. For which finite subgroups G ⊂ O(n) is Rn/G homeomorphic toRn.
One example when this holds is when G ⊂ U(n) is a finite subgroupgenerated by “complex reflections.” (A complex reflection is a linear auto-morphism of Cn with only one eigenvalue 6= 1 i.e., it is a rotation about
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a complex hyperplane.) For any complex reflection group G, Cn/G ∼= Cn.(This follows from the famous result that for such a G the ring of invariantpolynomials C[x1, . . . , xn]G is a polynomial ring on n variables.) IdentifyingCn with R2n we get R2n/G ∼= R2n. Another case where the answer to thequestion is affirmative is when G is the orientation-preserving subgroup of afinite group W generated by (real) reflections on Rn. We will see in Corol-lary 3.4 that Rn//W is a simplicial cone (which is homeomorphic to a halfspace). The orbifold Rn//G is the “double” discussed in Example 1.21 below.Hence, in this case we also have Rn?G ∼= Rn
After making these comments, I made the following conjecture. 1
Conjecture. Rn/G is homeomorphic to Rn if and only if either
(i) n = 2m and G is complex reflection group on Cm, or
(ii) G is the orientation-preserving subgroup of a real reflection group onRn.
Examples of orbifold coverings. Suppose X → |Q| is an ordinary cover-ing of topological spaces. Pullback the strata of Q to strata in X to obtainan orbifold Q′. (Here is a specific example: Q is RP 2 with one cone pointlabeled n. S2 → RP 2 is the double cover. The single cone point pulls backto two cone points in S2 labeled n.)
Example 1.21. Double |Q| along its boundary to get a 2-fold orbifold cov-ering Q′ → Q without codimension 1 strata. For example, if Q is a triangle,then Q′ is a 2-sphere with three cone points. As another example, if Q isthe nondevelopable orbifold pictured on the previous page (a 2-disk with onecorner reflector), then Q′ is the tear drop (a 2-sphere with one cone point).
Example 1.22. The n-fold branched cover of Q along a codimension 2 stra-tum labeled by the cyclic group of order n.
1There is an obvious counterexample to this conjecture: let G ⊂ SU(2) be the binarydodecahedral group of order 120. Then G acts freely on S3 and S3/G is Poincare’s homol-ogy 3-sphere. If we take the product of this representation with the trivial 1-dimensionalrepresentation we obtain a representation on R5 such that S4/G is the suspension of S3/G.It then follows from the Double Suspension Theorem of Cannon that R5/G is homeomor-phic to R5. The correct conjecture should be that this is the only counterexample.
Let Q denote the complement in |Q| of the strata of codimension ≥ 2(retain the mirrors on ∂|Q|). Choose a base point x0 in interior Q. Weare going to construct πorb1 (Q, x0) from π1(Q, x0) by adding generators andrelations.
New generators.
• For each component T of a codimension 2 stratum in interior of |Q|,choose a loop αT starting at x0 which makes a small loop around T .Let n(T ) be the order of the cyclic group labeling T .
• Suppose P is a codimension 2 stratum contained in M ∩N (so that Pis a corner reflector). Let m(P ) be the label on P (so that the dihedralgroup at P has order 2m(P )).
• For each mirror M and each homotopy class of paths γM from x0 to Mintroduce a new generator β(M,γM ).
Relations.
• [αT ]n(T ) = 1,
• [β(M,γM )]2 = 1, and
• ([β(M,γM )][β(N,γN )])m(P ) = 1,
Here P is a component of M ∩ N and γM and γN are homotopic as pathsfrom x0 to P .
2 Lecture 2: two-dimensional orbifolds
2.1 Orbifold Euler characteristics
We know what is meant by the Euler characteristic of a closed manifoldor finite CW complex (the alternating sum of the number of cells). A keyproperty is that it is multiplicative under finite covers: if M ′ → M is anm-fold cover, then
χ(M ′) = mχ(M).
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The Euler characteristic of an orbifold should be a rational number withsame multiplicative property, i.e., if M → Q is an m-fold cover and M is amanifold, then we should have χ(M) = mχorb(Q), i.e.,
χorb(Q) =1
mχ(M).
(“m-fold cover” means Card(p−1(generic point)) = m.)
The Euler characteristic of an orbifold. 2 Suppose Q is an orbifoldwhich is cellulated as a CW complex so that the local group is constant oneach open cell c. Let G(c) be the local group on c and |G(c)| denote its order.Define
χorb(Q) :=∑
cells c
(−1)dim c
|G(c)|.
Exercise 2.1. Suppose Γ y M properly, cocompactly, locally linearly andΓ′ ⊂ Γ is a subgroup of index m. Show
χorb(M//Γ′) = mχorb(M//Γ).
Alternate formula. Each stratum S of a compact orbifold Q is the interiorof a compact manifold with boundary S. Define e(S) := χ(S)−χ(∂S). Then
χorb(Q) =∑
strata S
e(S)
|G(S)|
Example 2.2. Suppose |Q| = D2 and Q has k mirrors and k corner reflectorslabeled m1, . . . ,mk. Then
χorb(Q) = 1− k
2+
(1
2m1
+ · · ·+ 1
2mk
)= 1− 1
2
∑i
(1− 1
mi
).
Example 2.3. Suppose |Q| = S2 and Q has l cone points labeled n1, . . . , nl.Then
χorb(Q) = 2− l +
(1
n1
+ · · ·+ 1
nl
)= 2−
∑i
(1− 1
ni
)(This is twice the previous example, as it should be.)
2In his lectures, Alejandro Adem gave a completely different definition of the “orbifoldEuler number, χorb(Q)”. For him, it is a certain integer which is defined using equivari-ant K-theory. Although this definition has been pushed by string theorists, the rationalnumber which I am using this terminology for goes back to Thurston’s 1976 course andbefore that to Satake.
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Example 2.4. (The general formula). Suppose |Q| is a surface with bound-ary and that Q has k corner reflectors labeled m1, . . . ,mk and l cone pointslabeled n1, . . . , nl. Then
χorb(Q) = χ(|Q|)− 1
2
k∑i=1
(1− 1
mi
)−
l∑i=1
(1− 1
ni
).
Remark 2.5. This formula shows that χorb(Q) ≤ χ(|Q|) with equality iffthere are no cone points or corner reflectors.
Notation 2.6. If a 2-dimensional orbifold has k corner reflectors which arelabeled m1, . . . ,mk and l cone points labeled n1, . . . , nl, we will denote thisby
(n1, . . . , nl;m1, . . . ,mk).
If ∂|Q| = ∅, then there can be no mirrors or corner reflectors and we simplywrite (n1, . . . , nl).
2.2 Classification of 2-dimensional orbifolds
Recall that closed surfaces are classified by orientability and Euler charac-teristic:
• χ(M2) > 0 =⇒ M2 = S2 or RP 2 (positive curvature).
• χ(M2) = 0 =⇒ M2 = T 2 or the Klein bottle (flat).
The idea is to classify orbifolds Q2 by their Euler characteristics. Sinceχorb( ) is multiplicative under finite covers, this will tell us which manifoldscan finitely cover a given orbifold. For example, if Q = S2//G, with G finite,then χorb(S2//G) > 0. Conversely, if Q is developable and χorb(Q) > 0, thenits universal cover is S2.
Exercise 2.7. List the 2-dimensional orbifolds Q with χorb(Q) ≥ 0. (In fact,I will do this exercise below.)
Since 1− (mi)−1 ≥ 1/2, we see that if k ≥ 4, then χorb(Q) ≤ 0 with equality
iff k = 4 and all mi = 2. Hence, if χorb(Q) > 0, then k ≤ 3.
More calculations. Suppose |Q| = D2 and k = 3 (so that Q is a triangle).Then
χorb(Q) =1
2(−1 + (m1)−1 + (m2)−1 + (m3)−1)
So, as (π/m1 + π/m2 + π/m3) is >, = or < π, χorb(Q) is, respectively, >, =or < 0. For χorb > 0, we see the only possibilities are: ( ; 2, 2,m), ( ; 2, 3, 3),( ; 2, 3, 4), ( ; 2, 3, 5). The last three correspond to the symmetry groups ofthe Platonic solids. For χorb(Q) = 0, the only possibilities are: ( ; 2, 3, 6),( ; 2, 4, 4) ( ; 3, 3, 3).
Making use of Remark 2.5, we do Exercise 2.7 below.
• The list of 2-dimensional spherical orbifolds is the list of finite sub-groups of O(3).
• Every 3-dimensional orbifold is locally isomorphic to the cone on oneof the spherical 2-orbifolds.
• For example, if |Q| = S2 with three cone points, (n1, n2, n3), thenCone(Q) has underlying space an open 3-disk. The three cone pointsyield three codimension 2 strata labeled m1, m2, m3 and the origin islabeled by the corresponding fintie subgroup of O(3).
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Flat orbifolds: χorb(Q) = 0: the 17 wallpaper groups.
Remark. In [14, pp. 103-115], Weyl emphasized the fact that there areexactly 17 discrete, cocompact subgroups of Isom(E2) up to conjugation inthe group of affine automorphisms. These 17 “wallpaper groups” are exactlythe orbifold fundamental groups of the orbifolds listed above.
χorb(Q) < 0: It turns out that all remaining 2-dimensional orbifolds aredevelopable and can be given a hyperbolic structure.
The triangular orbifolds, i.e., |Q| = D2; ( ;m1,m2,m3), with (m1)−1 +(m2)−1 +(m3)−1 < 1, have a unique hyperbolic structure (because hyperbolictriangles are determined, up to congruence, by their angles). The others havea positive-dimensional moduli space.
2.3 Spaces of constant curvature
In each dimension n, there are three simply connected spaces of constantcurvature: Sn (the sphere), En (Euclidean space) and Hn (hyperbolic space).
Definition 2.9. The hypersurface defined by 〈x, x〉 = −1 is a hyperboloidof two sheets. The component with xn+1 > 0 is Hn.
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xn+1
Figure 3: The quadratic form model of the hyperbolic plane
Definition 2.10. (The Riemannian metric on Hn). As in the case of asphere, given x ∈ Hn, TxHn = x⊥. Since 〈x, x〉 < 0, the restriction of 〈 , 〉to TxHn is positive definite. So, this defines a Riemannian metric on Hn. Itturns out this metric has constant secional curvature −1.
Geometric structures on orbifolds. Suppose G is a group of isometriesacting real analytically on a manifold X. (The only examples we will beconcerned with are Xn = Sn, En or Hn and G the full isometry group.) By
a (G,X)-structure we mean that each of the charts (U ,H, U, π) has U ⊂ X,that H is a finite subgp of G and the overlap maps (= compatibility maps)are required to be restrictions of isometries in G.
Convex polytopes in Xn. A hyperplane or half-space in Sn or Hn is the in-tersection of a linear hyperplane or half-space with the hypersurface. The unitnormal vector u to a hyperplane means that the hyperplane is the orthogonalcomplement, u⊥, of u (orthogonal wiith respect to the standard bilinear form,in the case of Sn, or the form 〈 , 〉, in the case of Hn). A half-space in Hn
bounded by the hyperplane u⊥ is a set of the form {x ∈ Hn | 〈u, x〉 ≥ 0} andsimilarly, for Sn. A convex polytope in Sn or Hn is a compact intersection ofa finite number of half-spaces.
Reflections in Sn and Hn. Suppose u is unit vector in Rn+1. Reflectionacross the hyperplane u⊥ (either in Rn+1 or Sn) is given by
x 7→ x− 2(x · u)u.
Similarly, suppose u ∈ Rn,1 satisfies 〈u, u〉 = 1. Reflection across the hyper-plane u⊥ in Hn is given by x 7→ x− 2〈x, u〉u.
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rL
rL´
π/m
Figure 4: The dihedral group of order 6
3 Lecture 3: reflection groups
3.1 Geometric reflection groups
Suppose K is a convex polytope in Xn (= Sn, En or Hn) such that if two codi-mension 1 faces have nonempty intersection, then the dihedral angle betweenthem has form π/m for some integer m ≥ 2. (This condition is familiar: itmeans that each codimension 2 face has the structure of a codimension 2 cor-ner reflector.) Let W be the subgroup of Isom(Xn) generated by reflectionsacross the codimension 1 faces of K.
Some basic facts:
• W is discrete and acts properly on Xn.
• K is a strict fundamental domain in the sense that the restriction toK of the orbit map, p : Xn → Xn/W , is a homeomorphism. It followsthat Xn//W ∼= K and hence, K can be given the structure of an orbifoldwith an Xn-structure.
(Neither fact is obvious.)
Example 3.1. A dihedral group is any group which is generated by twoinvolutions, call them s, t. It is determined up to isomorphism by the orderm of st (m is an integer ≥ 2 or the symbol ∞). Let Dm denote the dihedralgroup corresponding to m.
Example 3.2. For m 6=∞, Dm can be represented as the subgroup of O(2)which is generated by reflections across lines L, L′, making an angle of π/m.(See Figure 4.)
History and properties. 3
3In this paragraph I have relied on the Historical Note of [2, pp. 249-257].
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• In 1852 Mobius determined the finite subgroups of O(3) generated byisometric reflections on the 2-sphere.
• The fundamental domain for such a group on the 2-sphere was a spher-ical triangle with angles π
p, πq, πr, with p, q, r integers ≥ 2.
• Since the sum of the angles is > π, we have 1p
+ 1q
+ 1r> 1.
• For p ≥ q ≥ r, the only possibilities are: (p, 2, 2), for p ≥ 2, and (p, 3, 2)with p = 3, 4 or 5. (The last three cases are the symmetry groups ofthe Platonic solids.)
• Later work by Riemann and Schwarz showed there were discrete groupsof isometries of E2 or H2 generated by reflections across the edges oftriangles with angles integral submultiples of π. Poincare and Kleinproved similar results for polygons with more than three sides in H2.
In 2nd half of the 19th century work began on finite reflection groups onSn, n > 2, generalizing Mobius’ results for n = 2. It developed along twolines.
• Around 1850, Schlafli classified regular polytopes in Rn+1, n > 2. Thesymmetry group of such a polytope was a finite group generated byreflections and as in Mobius’ case, the projection of a fundamentaldomain to Sn was a spherical simplex with dihedral angles integralsubmultiples of π.
• Around 1890, Killing and E. Cartan classified complex semisimple Liealgebras in terms of their root systems. In 1925, Weyl showed thesymmetry group of such a root system was a finite reflection group.
• These two lines were united by Coxeter [4] in the 1930’s. He classifieddiscrete groups reflection groups on Sn or En.
Let K be a fundamental polytope for a geometric reflection group. ForSn, K is a simplex. For En, K is a product of simplices. For Hn there areother possibilities, e.g., a right-angled pentagon in H2 (see Figure [?]) or aright-angled dodecahedron in H3.
• Conversely, given a convex polytope K in Sn, En or Hn so that all dihe-dral angles have form π/integer, there is a discrete group W generatedby isometric reflections across the codimension 1 faces of K.
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Figure 5: Tessellation of hyperbolic plane by right-angled pentagons
• Let S be the set of reflections across the codimension 1 faces of K.For s, t ∈ S, let m(s, t) be the order of st. Then S generates W ,the faces corresponding to s and t intersect in a codimension 2 faceiff m(s, t) 6= ∞, and for s 6= t, the dihedral angle along that face isπ/m(s, t). Moreover,
• If m(, t) = 1 for s = t and is as defined above for s 6= t, then
〈S | (st)m(s,t), where (s, t) ∈ S × S〉
is a presentation for W .
Polytopes with nonobtuse dihedral angles.
Lemma 3.3. (Coxeter, [4]). Suppose K ⊂ Sn is an n-dimenional convexpolytope which is “proper” (meaning that it does not contain any pair ofantipodal points). Further suppose that whenever two codimension 1 facesintersect along a codimension 2 face, the dihedral angle is ≤ π/2. Then Kis a simplex.
A similar result holds for a polytope K ⊂ En which is not a product.
Corollary 3.4. The fundamental polytope for a spherical reflection group isa simplex.
Proof. For m an integer ≥ 2, we have π/m ≤ π/2.
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3.2 Simplicial Coxeter groups
3.2.1 The Gram matrix of a simplex in Xn
Suppose σn is a simplex in Xn. Let u0, . . . un be its inward pointing unitnormal vectors. (The ui lie in Rn+1, Rn or Rn,1 as Xn = Sn, En or Hn.) TheGram matrix, G, of σ is the symmetric (n+ 1)× (n+ 1) matrix (gij) definedby gij = ui · uj. G > 0 means the symmetric matrix G is positive definite.
Definition 3.5. A symmetric matrix G with 1’s on the diagonal is type
(1) if G > 0,
(0) if G is positive semidefinite with 1-dimensional kernel, each principalsubmatrix is> 0, and there is a vector v ∈ KerG with all its coordinates> 0,
(-1) if G has signature (n, 1) and each principal submatrix is > 0.
Linear algebra fact. The extra condition in type 0 (that KerG is spannedby a vector with positive coordinates) is automatic when G is indecomposableand has gij ≤ 0, for all i 6= j, i.e., when all dihedral angles are nonobtuse.(See [6, Lemma 6.3.7].)
Theorem 3.6. Suppose G is a symmetric (n+1)×(n+1) matrix with 1’s onthe diagonal. Let ε ∈ {+1, 0,−1}. Then G is the Gram matrix of a simplexσn ⊂ Xn
ε ⇐⇒ G is type ε.
Let Xnε is Sn, En, Hn as ε = +1, 0, −1.
Proof. For Sn: we can find basis vectors u0, . . . un in Rn+1, well-defined upto isometry, so that (ui · uj) = G. (This is because G > 0.) Since the uiform a basis, the half-spaces, ui ·x ≥ 0, intersect in a simplicial cone and theintersection of this with Sn is σn.
The proof for Hn is similar. The argument for En has additional compli-cations.
Suppose σn ⊂ Xn is a fundamental simplex for a geometric reflectongroup. Let {u0, . . . un} be the set of inward-pointing unit normal vectors.Then
ui · uj = − cos(π/mij)
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where (mij) is a symmetric matrix of posiive integers with 1’s on the diagonaland all off-diagonal entries ≥ 2. (The matrix (mij) is called the Coxetermatrix while the matrix (cos(π/mij)) is the associated cosine matrix.) Theformula above says:
Gram matrix = cosine matrix.
Suppose M = (mij) is a Coxeter matrix, i.e., a symmetric (n+ 1)× (n+ 1)matrix with 1’s on the diagonal and with off-diagonals ≥ 2 (sometimes weallow the off-diagonal mij to =∞, but not here).
Theorem 3.7. Let M be a Coxeter matrix as above and C its associatedcosine matrix (i.e., cij = − cos(π/mij)). Then there is a geometric refllectiongroup with fundamental simplex σn ⊂ Xn
ε ⇐⇒ C is type ε.
So, the problem of determining the geometric reflection groups with fun-damental polytope a simplex in Xn
ε becomes the problem of determining theCoxeter matrices M whose cosine matrix is type ε. This was done by Cox-eter, [4], for ε = 1 or 0 and by Lanner, [10], for ε = −1. The information ina Coxeter matrix is best encoded by its “Coxeter diagram.”
3.2.2 Coxeter diagrams
Associated to (W,S), there is a labeled graph Γ called its “Coxeter diagram.”Put Vert(Γ) := S. Connect distinct elements s, t by an edge iff m(s, t) 6= 2.Label the edge by m(s, t) if this is > 3 or =∞ and leave it unlabeled if it is= 3.
(W,S) is irreducible if Γ is connected. (The components of Γ give theirreducible factors of W .)
Figure 6 shows Coxeter’s classification from [4] of the irreducible spher-ical and cocompact Euclidean reflection groups. Figure 7 shows Lanner’sclassification from [10] of the hyperbolic reflection groups with fundamentalpolytope a simplex in Hn.
Exercise 3.8. Derive Lanner’s list in Figure 7 from Coxeter’s lists in Fig-ure 6.
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Spherical Diagrams Euclidean Diagrams
I (p) p2
H 53
H 54
F 44
ωA1∼
4 4B 2∼
6G2
∼
4F4
∼
E6
∼
E7
∼
E8
∼
E6
E7
E8
An
B 4n
Dn
nA∼
4B n∼
4 4Cn∼
Dn∼
Figure 6: Coxeter diagrams
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Hyperbolic Simplicial Diagrams
n = 2
with (p-1 + q-1 + r-1) < 1p q
r
n = 35
5 4
5 5
5
4
4 5 4
5 4
5 5
n = 4
5
5
5
5
4
5 4
Figure 7: Hyperboloic simplicial diagrams
3.3 More reflection groups
Recall Xn stands for Sn, En or Hn. Let K ⊂ Xn be a convex polytope withdihedral angles between codimension 1 faces of the form π/m, where m is aninteger ≥ 2 or the symbol ∞ (where π/∞ means the faces do not intersect).W the group generated by reflections across the codimension 1 faces of K.
Goal: Show W is discrete, acts properly on Xn and that K is an orbifoldwith geometric structure of an Xn-orbifold.
3.3.1 Generalities on abstract reflection groups
Suppose W is a group and S a set of involutions which generate it. For eachs, t ∈ S, let m(s, t) denote the order of st. (W,S) is a Coxeter system (andW is a Coxeter group) if the group defined by the presentation,
{generators} = S
{relations} = {(st)m(s,t)}, where (s, t) ∈ S × S, m(s, t) 6=∞,
is isomorphic to W (via the natural map).
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For each T ⊂ S, let WT denote the subgroup generated by T .
Definition 3.9. A mirror structure on a space X, indexed by a set S, isa family of closed subspaces {Xs}s∈S. The Xs are called mirrors. For eachx ∈ X, put S(x) := {s ∈ S | x ∈ Xs}.
Example 3.10. Suppose K is a convex polytope with its codimension 1faces indexed by S. For each s ∈ S, let Ks denote the face correspondingto s. This defines a mirror structure on K. S(x) is the set of faces whichcontain x. (In particular, if x is in the interior of K, then S(x) = ∅.)
The basic construction. Starting with a Coxeter system (W,S) and amirror structure {Xs}s∈S we are going to define a new space U(W,X) withW -action. The idea is to paste together copies of X, one for each elementof W . Each copy of X will be a fundamental domain and will be called a“chamber.”
Define an equivalence relation ∼ on W ×X by
(w, x) ∼ (w′, x′) ⇐⇒ x = x′ and wWS(x) = w′WS(x).
Here W has the discrete topology. (Recall that S(x) indexes the set of mirrorswhich contain x.) Put
U(W,X) = (W ×X)/ ∼ .
To simplify notation, write U for U(W,X). Denote the image of (w, x) in Uby [w, x].
Some properties of the construction.
• W y U via u[w, x] = [uw, x]. The isotropy subgroup at [w, x] iswWS(x)w
−1.
• We can identify X with the image of 1×X in U . X is a strict funda-mental domain for the W -action in the sense that the restriction of theorbit map U → U/W to X is a homeomorphism (i.e., U/W = X).
• W y U properly ⇐⇒ X is Hausdorff and each WS(x) is finite (i.e.,⋂s∈T Xs = ∅, whenever |WT | =∞).
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Universal property. Suppose W y Z and f : X → Z is a map so that forall s ∈ S, f(Xs) ⊂ Zs. (Zs denotes the fixed set of s on Z.) Then there isa unique extension to a W -equivariant map f : U(W,X)→ Z. (In fact, f isdefined by f([w, x]) = wf(x).)
Exercise 3.11. Prove the above properties hold.
3.3.2 Geometric reflection groups, again
The set up:
• K is a convex polytope in Xn (= Sn, En or Hn). S is the set of reflectionsacross the codimension 1 faces of K. The face corresponding to s isdenoted by Ks.
• If Ks ∩ Kt 6= ∅, then it is a codimension 2 stratum and the dihedralangle is π/m(s, t), where m(s, t) is some integer ≥ 2. (We know thisimplies K is a simple polytope.) If Ks∩Kt = ∅, then put m(s, t) =∞.
• Let W ⊂ Isom(Xn) be the subgroup generated by S.
• Let W be the group defined by the presentation corresponding to the(S×S) Coxeter matrix, (m(s, t)). It turns out that (W,S) is a Coxetersystem. (There is something to prove here, namely, that the order of stis = m(s, t) rather than that it just divides m(s, t).) Let p : W → Wbe the natural surjection.
By the universal property, the inclusion ι : K ↪→ Xn induces a W -equivariantmap ι : U(W,X)→ Xn.
Theorem 3.12. ι : U(W,K)→ Xn is a W -equivariant homeomorphism.
Some consequences:
• p : W → W is an isomorphism
• W is discrete and acts properly on Xn
• K is a strict fundamental domain for the action on Xn (i.e., Xn/W =K).
• U(W,K) is a manifold (because Xn is a manifold).
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• K is an Xn-orbifold (because it is identified with Xn/W ).
• If W ′ y Rn as a finite linear group, then Rn//W is isomorphic to thefundamental simplicial cone.
Sketch of proof of the theorem. The proof is by induction on the dimensionn. A neighborhood of a point in K looks like the cone over the suspension,σ, of a spherical simplex. By induction, U(WT , σ) = Sn−1 (where WT is thefinite Coxeter group corresponding to σ). Since a neighorhood inK is an openXn-cone over σ, it follows that ι : U(W,K)→ Xn is a local homeomorphismand a covering projection and that U(W,K) has the structure of an Xn-manifold. Since Xn is simply connected, the covering projection ι ust be ahomeomorphism. (The case Xn = S1 is handled separately.)
4 Lecture 4: 3-dimensional hyperbolic reflec-
tion groups
4.1 Andreev’s Theorem
A geometric reflection group on Sn, En or Hn is determined by its funda-mental polytope. In the spherical case the fundamental polytope must bea simplex and in the Euclidean case it must be a a product of simplices.Furthermore, all the possibilities for these simplices are listed in Figure 6.So, there is nothing more to said in the spherical and Euclidean cases.
In the hyperbolic case we know what happens in dimension 2: the fun-damental polygon can be an k-gon for any k ≥ 3 and almost any assignmentof angles can be realized by a hyperbolic polygon (there are a few exceptionswhen k = 3 or 4). What happens in dimension 3?
There is a beautiful theorem due to Andreev, which gives a completeanswer. Roughly, it says given a simple polytope K, for it to be the fundpolytope of a hyperbolic reflection group,
• there is no restriction on its combinatorial type
• subject to the condition that the group at each vertex be finite, almostany assignment of dihedral angles to the edges of K can be realized(provided a few simple inequalities hold).
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In contrast to dimension 2, the 3-dimensional hyperbolic polytope is uniquelydetermined, up to isometry, by its dihedral angles – the moduli space is apoint.
Remark. By a theorem of Vinberg, hyperbolic examples do not exist indimensions ≥ 30.
Theorem 4.1 (Thurston’s Conjecture, Perelman’s Theorem). A closed, de-velopable 3-orbifold Q3 with infinite πorb1 admits a hyperbolic structure iff itsatisfies the following two conditions:
(i) Every embedded 2-dimensional spherical suborbifold bounds a quotientof a 3-ball in Q3. (This condition implies Q3 is aspherical.)
(ii) Z× Z 6⊂ πorb1 (Q3).
A 2-dimensional suborbifold of Q3 is incompressible if the inclusion intoQ3 induces an injection on πorb1 ( ). The orbifold Q3 is Haken if it does notcontain any nondevelopable 2-dimensional suborbifolds, if every spherical 2-dimensional suborbifold bounds the quotient of a 3-ball by a finite lineargroup and if it contains an incompressible 2-dimensional Euclidean or hyper-bolic orbififold.
Proposition 4.2. ([13, Prop. 13.5]). An orbifold with underlying space a3-disk and with no singular points in its interior (called a “reflectofold” inSubsection 5.1) is Haken iff it is neither a tetrahedron nor the product of atriangular spherical orbifold with [0, 1] (i.e., a triangular prism).
In the late 1970’s Thurston proved his conjecture for Haken manifolds ororbifolds. This can be stated as follows.
Theorem 4.3. (Thurston ∼ 1977). A 3-dimensional Haken orbifold Q3
admits a hyperbolic structure iff it has no incompressible 2-dimensional Eu-clidean suborbifolds (i.e., Q3 is “atoroidal”).
Combining this with Proposition 4.2 we get Corollary ?? below as a spe-cial case. This had been proved several years earlier by Andreev as a corollaryto the following theorem about convex polytopes in H3.
Theorem 4.4. (Andreev ∼ 1967, see [1, 11]). Suppose K is (the combina-torial type of) a simple 3-dimensional polytope, different from a tetrahedron.Let E be its edge set and θ : E → (0, π/2] any function. Then (K, θ) can berealized as a convex polytope in H3 with dihedral angles as prescribed by θ ifand only if the following conditions hold:
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(i) At each vertex, the angles at the three edges e1, e2, e3 which meet theresatisfy θ(e1) + θ(e2) + θ(e3) > π.
(ii) If three faces intersect pairwise but do not have a common vertex, thenthe angles at the three edges of intersection satisfy θ(e1)+θ(e2)+θ(e3) <π.
(iii) Four faces cannot intersect cyclically with all four angles = π/2 unlesstwo of the opposite faces also intersect.
(iv) If K is a triangular prism the angles along base and top cannot all beπ/2.
Moreover, when (K, θ) is realizable, it is unique up to an isometry of H3.
Corollary 4.5. Suppose K is (the combinatorial type of) a simple 3-polytope,different from a tetrahedron, that {Fs}s∈S is its set of codimension 1 faces andthat est is the edge Fs∩Ft (when Fs∩Ft 6= ∅). Given an angle assignment θ :E → (0, π/2], with θ(est) = π/m(s, t) and m(s, t) an integer ≥ 2, then (K, θ)is a hyperbolic orbifold iff the θ(est) satisfy Andreev’s Conditions. Moreover,the geometric reflection group W is unique up to conjugation in Isom(H3).
Remark. The condition that K is not a tetrahedron and Andreev’s Condi-tion (iv) deal with the case when the orbifold K is not Haken.
Examples 4.6. Here are some hyperbolic orbifolds:
• K is a dodecahedron with all dihedral angles equal to π/2.
• K is a cube with disjoint edges in different directions labeled by integers> 2 and all other edges labeled 2.
Exercise 4.7. Make up your own examples.
The dual form of Andreev’s Theorem. Let L be the triangulation of S2
dual to ∂K.
Vert(L)←→ Face(K)
Edge(L)←→ Edge(K)
{2-simplices in L} ←→ Vert(K)
Input data. Suppose we are given θ : Edge(L) → (0, π/2]. The conditionthat K has a spherical link at each vertex is that if e1, e2, e3 are the edges ofa triangle, then θ(e1) + θ(e2) + θ(e3) > π.
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Theorem 4.8. (Dual form of Andreev’s Theorem). Suppose L is a triangu-lation of S2 and L 6= ∂∆3. Let θ : Edge(L)→ (0, π/2] be any function. Thenthe dual polytope K can be realized as convex polytope inH3 with prescribeddihedral angles iff the following conditions hold:
(i) If e1, e2, e3 are the edges of any triangle, then θ(e1) + θ(e2) + θ(e3) > π.
(ii) If e1, e2, e3 are the edges of a 3-circuit 6= ∂∆2, then θ(e1)+θ(e2)+θ(e3) <π.
(iii) If e1, e2, e3, e4 are the edges of a 4-circuit which is 6= to boundary ofunion of two adjacent triangles, then all four θ(ei) cannot = π/2.
(iv) If L is suspension of ∂∆2, then all “vertical” edges cannot have θ(ei) =π/2.
A dimension count. Given a convex 3-dimensional polytope K, Andreev’sTheorem asserts that a certain map Θ from the space C(K) of isometryclasses convex polyhedra of the same combinatorial type as K to a certainsubset A(K) ⊂ RE is a homeomorphism (where E := Edge(K) and whereA(K) is the convex subset defined by Andreev’s inequalities).
Let’s compute dimC(K). For each F ∈ Face(K), let uF ∈ S2,1 be theinward-pointing unit normal vector to F . (Here S2,1 := {x ∈ R3,1 | 〈x, x〉 =1}.) The (uF )F∈Face(K) determine K (since K is the intersection of the half-spaces determined by the uF ). The assumption that K is simple means thatthe hyperbolic hyperplanes normal to the uF intersect in general position.So, a slight perturbation of the uF will not change the combinatorial type ofK. That is to say, the subset of Face(K)-tuples (uF ) which define a polytopecombinatorially isomorphic to K is an open subset Y of (S2,1)Face(K).
• Let f = Card(Face(K)), e = Card(Edge(K), v = Card(Vert(K).
• Isom(H3) = O(3, 1), dim(O(3, 1) = 6, and dim S2,1 = 3.
• So, dimC(K) = 3f − 6.
Since f − e + v = 2, we have 3f − 6 = 3e − 3v. Since three edges meetat each vertex, we have 3v = 2e. Hence, 3f − 6 = 3e − 3v = e. So,Θ : C(K)→ A(K) ⊂ RE is a map between manifolds with boundary of thesame dimension.
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4.2 3-dimensional orbifolds
Recall the list of 2-dimensional spherical orbifolds from Subsection 2.2:
The local models for 3-dimensional orbifolds are cones on any one of theabove.
For example, if |Q2| = S2 with (n, n), then the 3-dimensional model isD3 with an interval of cone points labeled n. Quotients of n-fold branchedcovers of knots or links in S3 (or any other 3-manifold) have this form.
Example 4.9. (A flat orbifold). Consider the 3 families of lines in E3 of theform (t, n,m+ 1
2), (m+ 1
2, t, n) and (n,m+ 1
2, t), where t ∈ R and n,m ∈ Z.
Let Γ be the subgroup of Isom(E3) generated by rotation by π about eachof these lines. A fundamental domain is the unit cube. The orbifold E3//Γis obtained by “folding up” the cube to get the 3-sphere. The image of thelines (= the singular set) are 3 circles in S3 each labeled by 2 (meaning C2,the cyclic group of order 2). These 3 circles form the Borromean rings. (See[13] for pictures of the folding up process.)
Example 4.10. Suppose Q is an orbifold with underlying space S3, withsingular set the Borromean rings and with the components of the singularset labeled by cyclic groups of order p, q and r. I showed in my lecturehow to use Andreev’s theorem to show that this orbifold admits a hyperbolicstructure iff all three integers are > 2. The proof uses the second example inExamples 4.6.
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Figure 8: A nondevelopable reflectofold
5 Lecture 5: aspherical orbifolds
5.1 Reflectofolds
Definition 5.1. An n-dimensional orbifold Q is a reflectofold 4 if is locallymodeled on finite linear reflection groups acting on Rn.
If W acts on Rn as a finite reflection group, then Rn/W is a simplicialcone, i.e., up to linear isomorphism it looks like [0,∞)n. It follows that theunderlying space of a reflectofold Q is a manifold with corners. Conversely,to give a manifold with corners the structure of a reflectofold, essentially allwe need to do is label its codimension 2 strata by integers ≥ 2 in such a waythat the strata of higher codimension correspond to finite Coxeter groups(which are listed in Figure 6).
It follows from the description of πorb1 (Q) in Subsection 1.3 that πorb1 (Q) isgenerated by reflections if and only if π1(|Q|) = 1. (Here “reflection” meansan involution with codimension 1 fixed set.) Henceforth, let’s assume this(that |Q| is simply connected).
If Q is developable, then any codimension 2 stratum is contained in theclosures of two distinct codimension 1 strata. Otherwise, we would have anondevelopable suborbifold pictured in Figure 8. Similarly, developabilityimplies that if intersection of two codimension 1 strata contains two distinctcodimension 1 strata, then they must be labeled by the same integer.
4When I introduced this term in my lecture I suggested that, as in Thurson’s class, weshould have an election to name the concept. Lizhen Ji was enthusiastic about this idea;however, in the end I didn’t implement it.
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5.2 Asphericity
Definition 5.2. An orbifold is aspherical if its universal cover is a con-tractible manifold.
One might ask why, in the above definition, we require the universalcover to be a contractible manifold rather than just a contractible orbifold.(A contractible orbifold is a simply connected orbifold all of whose higherhomotopy groups also vanish. This definiton does not automatically implythat the orbifold is developable.) In fact, in the next questin we ask if itmakes any difference which condition is required.
Question. Is it true that every contractible orbifold is developable?
Remark. I think the question has an affirmative answer, but I have neverseen it written down.
Remark. A 2-dimensional orbifold Q2 is aspherical ⇐⇒ χorb(Q2) ≤ 0.
My favorite conjecture.
Conjecture. (Hopf, Chern, Thurston). Suppose Q2n is a closed asphericalorbifold. Then (−1)nχorb(Q2n) ≥ 0.
Hopf and Chern made this conjecture for nonpositively curve manifolds(I believe they thought it might follow from the Gauss-Bonnet Theorem) andThurston extended it to aspherical manifolds (at least in the 4-dimensionalcase). For much more about this conjecture in the case of aspherical reflecto-folds, see [6].
The set up. Let Q be a reflectofold. Denote the underlying space by K(instead of |Q|). Let S index the set of mirrors (= {codimension 1 strata}).Ks denotes the closed mirror corresponding to s. Let m(s, t) be the label onthe codimension 2 strata in Ks ∩Kt. Put m(s, t) = ∞ if Ks ∩Kt = ∅. Let(W,S) be the Coxeter system defined by the presentation (i.e., W = πorb1 (Q)).For each T ⊂ S, let WT denote the subgroup generated by T . The subsetT is spherical if WT is finite. Let S be the set of spherical subsets of S,partially ordered by inclusion. (N.B. ∅ ∈ S.) Put
KT =⋂s∈T
Ks.
Since Q is an orbifold, whenever KT 6= ∅, we must have WT ∈ S.
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Theorem 5.3. The reflectofold Q is aspherical iff the following conditionshold:
(i) KT 6= ∅ ⇐⇒ T ∈ S (i.e., when WT is finite).
(ii) For each T ∈ S, KT is acyclic (i.e., H∗(KT ) = 0).
(Notes: K∅ = K; also, when K is simply connected and acyclic, it is con-tractible.)
The first condition means that the combinatorics of intersections of mir-rors is determined by (W,S). It is the analog of Andreev’s Conditions (with-out the atoroidal condition), cf. Theorem 4.8. The second condition says thatthe manifold with corners K “looks like” a convex polytope up to homology.We elucidate these points below.
Definition 5.4. The nerve of the mirror structure {Ks}s∈S on K is anabstract simplicial complex L′ defined as follows: its vertex set, Vert(L′),is S and a nonempty subset T of S is the vertex set of a simplex in L′ iffKT 6= ∅.
If L′′ is any simplicial complex with Vert(L′′) = S, write S(L′′) for theposet of vertex sets of simplices in L′′. If σ is a simplex of L′′ with vertex setT , let Lk(σ, L′′) denote the abstract simplicial complex corresponding to theposet S(L′′)>T . (Lk(σ, L′′) is called the link of σ in L′′.) Given two topologicalspaces X and Y , write X ∼ Y to mean that H∗(X; Z) ∼= H∗(Y ; Z). IfdimK = n, then, by standard arguments in algebraic topology, condition(ii) of Theorem 5.3 means that
L′ ∼ Sn−1 and Lk(σ, L′) ∼ Sn−1−dimσ. (5.1)
for all simplices σ in L′ (cf., [5] or [6, §8.2]). In the case where K is a convexpolytope, L′ is the boundary of the dual polytope, i.e., L′ is dual to ∂K (cf.the last part of Subsection 4.1).
Definition 5.5. Suppose (W,S) is a Coxeter system. The elements of Swhich are 6= ∅ are the simplices of an abstract simplicial complex, denotedby L(W,S) (or more simply, by L) and called the nerve of (W,S). Moreprecisely, Vert(L) = S and a nonempty subset T of S is the vertex set of asimplex in L iff T is spherical.
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The fact that K is the underlying space of an orbifold means that L′ ⊂ L(i.e., all local groups are finite). Condition (i) of Theorem 5.3 is that L′ = L.By Condition (ii), L satisfies (5.1).
Before sketching the proof of Theorem 5.3 we discuss the following twoquestions:
(A) How do you produce a large number of examples of Coxeter systems(W,S) with L(W,S) satisfying (5.1)?
(B) How do you recover K from L?
More generally, how do we find Coxeter system (W,S) with nerve a givenfinite simplicial complex J? We should start as follows. Put S = Vert(J).Label each edge {s, t} by an integer m(s, t) ≥ 2. This defines the Coxetersystem (W,S). The condition that we need to get an orbifold is that wheneverT is the vertex set of a simplex of J , then T ∈ S. Condition (i) of Theorem 5.3(an analog of Andreev’s Theorem) is the converse: whenever T ∈ S, thenT is the vertex set of a simplex in J . We will see below that when all them(s, t)’s are 2 or ∞ these conditions are easy to decide.
Definition 5.6. A simplicial complex J is a flag complex if T is any finite,nonempty collection of vertices which are pairwise connected by edges, thenT spans a simplex of J
Remark 5.7. In [8] Gromov uses the terminology that J satisfies the “no∆ condition” for this concept. I once used the terminology that J is “deter-mined by its 1-skeleton” for the same notion. Combinatorialists call such aJ a “clique complex”.
Examples 5.8.
• If J is a k-gon (i.e., a triangulation of S1 into k edges), then J is a flagcomplex iff k > 3.
• The barycentric subdivision of any simplicial complex (or, in fact, ofany cell complex) is a flag complex.
The second of Examples 5.8 shows that the condition of being a flagcomplex does not restrict the topological type of J – it can be any polyhedron.
Definition 5.9. A Coxeter system (W,S) is right-angled if for each s 6= t,m(s, t) is either 2 or ∞.
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Since the nerve of any right-angled (W,S) is obviously a flag complex,the second of Examples 5.8 yields the following answer to Question (A).
Proposition 5.10. The barycentric subdivision of any finite cell complexoccurs as the nerve of a right-angled Coxeter system.
Reconstructing K. Now suppose that L (= L′) is a PL triangulationof Sn−1. Let K = Dn (= Cone(Sn−1) and identify ∂K with L. We wantto find a mirror structure on K dual to L. The construction is the usualone for defining the dual cell structure on a manifold. For each vertex s ofL, let Ks be the closed star of s in the barycentric subdivision, bL. Thus,Ks = Cone(bLk(s, L)). For each T ∈ S, we then have KT =
⋂s∈T Ks =
Cone b(Lk(σT , L)), where σT is the simplex in L corresponding to T . Theassumption that the triangulation is PL means that each Lk(σT , L) is asphere (= Sn−1−dimσT ); so, each KT is a cell.
Exactly the same construction works when L is a PL triangulation ofa homology sphere (that is, a closed PL manifold with the same homologyas Sn−1), except that instead of being a disk, K is a compact contractiblemanifold with boundary L. (This uses the fact that any homology sphere Lis topolgically the boundary of a contractible 4-manifold. This fact followsfrom surgery theory when dimL > 3 and and is due to Freedman whendimL = 3.) In general, when L is only required to satisfy (5.1), one mustrepeatedly apply this step of replacing Cone(bLk(σ, L)) by a contractiblemanifold bounded by a contractible manifold (see [5]).
5.3 Proof of the asphericity theorem
For each w ∈ W , define the following subset of S:
In(w) := {s ∈ S | l(ws) < l(w)}.
(l(w) is the word length of w with respect to the generating set S.)The following lemma in the theory of Coxeter groups is key to the proof
of Theorem 5.3.
Lemma 5.11. (See [6, Lemma 4.7.2]). For each w ∈ W , In(W ) is a sphericalsubset of S.
Sketch of proof of Theorem 5.3. The universal cover of the reflectofold Q isthe manifold U(W,K), which we denote simply by U . The manifold U is
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contractible if and only if it is simply connected and acyclic. We will derivenecessary and sufficient conditions for this to hold.
Order the elemnts of W : w1, w2, . . . wk . . . , in any fashion so that l(wk) ≤l(wk+1). Let Pk denote the union of the first k “chambers” in U , i.e.,
Pk := w1K ∪ · · · ∪ wkK.
We propose to study the exact sequence of the pair (Pk+1, Pk) in homology.To simplify notation, put w = wk+1. By excision,
H∗(Pk+1, Pk) = H∗(wK,wKIn(w)) = H∗(K,KIn(w)),
where for any subset T ⊂ S,
KT :=⋃s∈T
Ks.
So, the long exact sequence of the pair becomes
· · · → H∗(Pk)→ H∗(Pk+1)→ H∗(K,KIn(w))→ . . . .
It is not hard to see that there is a splitting, H∗(K,KIn(w)) → H∗(Pk+1), of
the right hand map defined by multiplication by wh∈(w)w−1 ∈ ZW , where
for any T ∈ S, hT is the element in the group ring ZWT defined by hT :=∑u∈WT
(−1)l(u)u. Hence,
H∗(Pk+1) ∼= H∗(Pk)⊕H∗(K,KIn(w)) and therefore,
H∗(U) ∼=∞⊕k=1
H∗(K,KIn(wk))
If AT denotes the free abelian group on {w ∈ W | In(w) = T}, then theabove formula can be rewritten as
H∗(U) ∼=⊕T∈S
H∗(K,KT )⊗ AT . (5.2)
From (5.2) we see that H∗(U) = 0 iff H∗(K,KT ) = 0 for all T ∈ S. Standard
arguments using Mayer-Vietoris sequences (or the Mayer-Vietoris spectralsequence) show that these terms all vanish iff for all T ∈ S, the intersectionKT is acyclic (this includes the statement that K is acyclic. (See [6, §8.2].)
A similar argument using van Kampen’s Theorem applied to Pk+1 =Pk ∪K shows that U is simply connected iff K is simply connected, each Ks
is connected and for each {s, t} ∈ S, K{s,t} 6= ∅.
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5.4 Aspherical orbifolds not covered by Euclidean space
A noncompact space X is simply connected at infinity if given any compactsubset C there is a larger compact subset D so that any loop in X − D isnull homotopic in X − C. In favorable circumstances the inverse system offundamental groups {π1(X −C)}, where C ranges over all compact subsets,has a well-defined inverse limit, π∞1 (X), called the fundamental group atinfinity. If X is simply connected at infinity, then π∞1 (X) is trivial. (See [7]for the basic facts about the concepts in this paragraph.)
Theorem 5.12. (Stallings, Freedman, Perelman). A contractible n-manifoldis homeomorphic to Rn iff it is simply connected at infinity.
(Stallings proved this in dimensions ≥ 5, Freedman in dimension 4 and indimension 3, I believe it follows from Perelman’s proof of the Poincare Con-jecture.)
For some time it was an open problem if the universal cover of a closed,aspherical manifold had to be homeomorphic to Euclidean space. Of course,the issue was not the existence of exotic (i.e., not simply connected at infinity)contractible manifolds but whether such an exotic contractible manifold couldadmit a cocompact transformation group. This was resolved in [5] by usingthe techniques of this section.
Let L be a triangulation of a homology (n− 1)-sphere as a flag complex.Label its edges by 2 and let (W,S) be the associated right-angled Coxetergroup with nerve L. Let K be a contractible manifold with ∂K = L. Asexplained above, we can put the dual cell structure on ∂K to give K thestructure of a manifold with corners and hence, the structure of a reflectofoldQ. The claim is that if n > 2 and Ln−1 is not simply connected, then thecontractible manifold U(W,K) is not simply connected at infinity. As before,
let Pk be the union of the first k chambers and let◦P k be its interior. The
argument goes as follows.
• Since Pk is obtained by gluing on a copy of K to Pk−1 along an (n−1)-disk in its boundary, it follows that Pk is a contractible manifold withboundary and that its boundary is the connected sum of k copies ofπ1(∂K). Hence, π1(∂(Pk)) is the free product of k copies of π1(∂k).
• For a similar reason one U −◦P k is homotopy equivalent to ∂Pk.
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• Hence, π∞1 (U) is the inverse limit, lim−→(π1(L) ∗ · · · ∗ π1(L)). In otherwords, it is the “projective free product” of copies of π1(L). In partic-ular, it is nontrivial whenever π1(L) 6= 1.
The above is a sketch of the proof of the following result.
Theorem 5.13. ([5]) For each n ≥ 4 there are closed, aspherical n-dimensionalorbifolds with universal cover not homeomorphic to Rn.
Since Coxeter groups have faithful linear representations (cf. [2]), Sel-berg’s Lemma implies that they are virtually torsion-free. So, there is atorsion-free subgroup Γ ⊂ W which then necessarily acts freely on U . Hence,M = U/Γ is a closed, aspherical manifold. Thus, the previous theorem hasthe following corollary.
Corollary 5.14. ([5]) For each n ≥ 4, there are closed, aspherical n-dimensionalmanifolds with universal cover not homeomorphic to Rn.
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[8] M. Gromov, Hyperbolic groups in Essays in Group Theory, edited by S.M. Gersten, M.S.R.I. Publ. 8, Springer, New York, 1987, pp. 75-264.
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